Android app updates fail to download
I had two Android devices (running different major releases of Android) that could not update any apps. All of the online suggestions I found for this problem were device-specific -- clearing app caches, doing a factory reset, etc. I knew that both devices started having problems at the same time, so a device-specific solution didn't seem likely (and some of those solutions were pretty drastic, especially considering I wasn't confident they would work). I'm glad I didn't try any of the suggestions that had major impact, because none of them would have worked. As I suspected, the problem wasn't on the devices.
The devices in question were a phone and a tablet. The tablet does not have cell service, it is WiFi-only. Out of curiosity, I tried turning the phone's WiFi off and attempted an app update using my mobile data plan. And it worked! This gave me confidence in my analysis that it wasn't likely a device-specific problem. But what could it be? Well, I remembered that I'd recently changed my router.
To make a long story short, the problem was related to the DNS server configuration on the router (using DD-WRT). I can't explain why I was getting the results I was, but something about the combination of the new router, the newer build of DD-WRT, the DNS settings copied from the old router, and Android and/or Play Services did not get along. By changing the router's static DNS server addresses, the problem was fixed. Both devices can download and install updates normally now.
So, if you're having problems getting Android apps to update, especially if they get stuck on the download phase (i.e., they don't download at all), don't do a factory reset of your phone until you've tried the lower-impact option of changing the DNS settings on your router.